A gas is filled in a container of volume V at `121^(@)C`. To what temperature should it be heated in order that `(1)/(4)` th of the gas may escape out of the vessel?

Explanation

To solve the problem, we will follow these steps:

### Step 1: Convert the initial temperature from Celsius to Kelvin
The initial temperature of the gas is given as \(121^\circ C\). To convert this to Kelvin, we use the formula:
\[ T(K) = T(°C) + 273 \]
Substituting the given value:
\[ T_1 = 121 + 273 = 394 \, K \]

### Step 2: Write the ideal gas equation for initial conditions
Using the ideal gas equation \(PV = nRT\), we can write the equation for the initial state of the gas:
\[ PV = nR(394) \]
Where:
- \(P\) = pressure
- \(V\) = volume
- \(n\) = number of moles
- \(R\) = ideal gas constant

### Step 3: Determine the final number of moles after gas escapes
Since one-fourth of the gas escapes, the remaining number of moles is:
\[ n_f = n - \frac{n}{4} = \frac{3n}{4} \]

### Step 4: Write the ideal gas equation for final conditions
For the final state, the volume remains the same (V), but the number of moles is now \(n_f = \frac{3n}{4}\). The equation becomes:
\[ PV = \left(\frac{3n}{4}\right)RT_f \]

### Step 5: Divide the initial and final equations
Dividing the initial equation by the final equation gives:
\[ \frac{PV}{PV} = \frac{nR(394)}{\left(\frac{3n}{4}\right)RT_f} \]
This simplifies to:
\[ 1 = \frac{394}{\frac{3}{4}T_f} \]

### Step 6: Solve for the final temperature \(T_f\)
Rearranging the equation to solve for \(T_f\):
\[ T_f = \frac{394 \times \frac{3}{4}}{1} = \frac{394 \times 4}{3} \]
Calculating \(T_f\):
\[ T_f = \frac{1576}{3} \approx 525.33 \, K \]

### Step 7: Convert the final temperature back to Celsius
To convert the final temperature from Kelvin to Celsius, we use:
\[ T(°C) = T(K) - 273 \]
Substituting the value:
\[ T_f(°C) = 525.33 - 273 \approx 252.33 \, °C \]

### Final Answer
The final temperature to which the gas should be heated is approximately:
\[ \boxed{252.33 \, °C} \]

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